it ed 
In the same way we find the increase in consequence of the 
systems passing the other limits, — taking into account that all 
eye . . da, rr . . 
quantities of the form — „are zero. — Faking the sum of all these 
a, at 
1 
partial increases and dividing by |(d¢/,)| [(da@,)| we find: 
OD OAD GFN er OD dae . 
Cy alae ale ene a ay 
OD (OD df, “(0D das] 4D F 
a ae ice ll ET PRG, 
0D 
Here ee represents the fluction of the density for a phasis whose 
limits are constant, = for a phasis whose limits partake of the motion 
Ll 
of the systems of the ensemble. 
So the density proves to be constant for a phasis, partaking of 
the motion of the systems, and as, of course, the systems can never 
pass the limits of an extension-in-phase, when these limits move with 
the systems, the total number of systems within every extension-in- 
phase, i.e. D{(df,)] [(de,)] remains constant, and so also [(d¢7,)] {(de,)]. 
This proof of the laws of conservation of density-in-phase and of 
extension-in-phase quite agrees with that one given by GiBBs. Im 
our case, however, we have still to pay attention to one circumstance. 
is OD fe 
In calculating ra A) \[(da,)|, we have assumed, that this number 
is the sum of the numbers of systems passing the different limits. 
This comes to the same as to say that no system will pass more 
than one of the limits during the time dé, or at least, that the number 
of the systems that pass more than one limit is so small, that it may 
be neglected. In the proof of GriBBs we may assume, that this is 
dy 
really the case, provided we take df so small, that ms dt is small 
compared with dy (where g represents one of the generalized coor- 
dinates, and (/q one of the dimensions of an element of extension-in- 
Y phase). For our case however this proof is 
incomplete. Be 7 and s two adjacent elements 
of space, then [f,] and [fs], [@,] and [a;) are 
no independent quantities, but they must be 
A approximately equal, as [7] and [a] vary only 
fluently from point to point. 
In order to investigate the consequences of 
O x this circumstance we imagine an ensemble of 
